## Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two (1997)

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Venue: | Journal of Cryptology |

Citations: | 33 - 0 self |

### BibTeX

@ARTICLE{Müller97fastmultiplication,

author = {Volker Müller and Volker Muller and Technische Hochschule Darmstadt and Fachbereich Informatik},

title = {Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two},

journal = {Journal of Cryptology},

year = {1997},

volume = {11},

pages = {219--234}

}

### Years of Citing Articles

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### Abstract

We discuss new algorithms for multiplying points on elliptic curves over small finite fields of characteristic two. This algorithm is an extension of previous results by Koblitz, Meier and Staffelbach. Practical timings show that the new methods can give a running time improvement of up to 50% compared to the ordinary binary algorithm for multiplication. Finally, we present a table of elliptic curves, which are well suited for elliptic curve public key cryptosystems, and for which the new algorithm can be used. 1 Introduction Elliptic curves over finite fields have gained a lot of attention in public key cryptography in recent years ([4], [10]). For practical reasons, elliptic curves over fields of characteristic two are of special interest. Diffie-Hellman type cryptosystems using elliptic curves over IF 2 155 were implemented and compared to RSA (see [12]). The most time consuming operation of these cryptosystems is multiplication of a point on the elliptic curve with an integer, wh...

### Citations

697 |
Elliptic curve cryptosystems
- Koblitz
- 1987
(Show Context)
Citation Context ...rve public key cryptosystems, and for which the new algorithm can be used. 1 Introduction Elliptic curves over finite fields have gained a lot of attention in public key cryptography in recent years (=-=[4]-=-, [10]). For practical reasons, elliptic curves over fields of characteristic two are of special interest. Diffie-Hellman type cryptosystems using elliptic curves over IF 2 155 were implemented and co... |

531 |
Use of elliptic curves in cryptography
- Miller
- 1986
(Show Context)
Citation Context ...ublic key cryptosystems, and for which the new algorithm can be used. 1 Introduction Elliptic curves over finite fields have gained a lot of attention in public key cryptography in recent years ([4], =-=[10]-=-). For practical reasons, elliptic curves over fields of characteristic two are of special interest. Diffie-Hellman type cryptosystems using elliptic curves over IF 2 155 were implemented and compared... |

306 | An improved algorithm for computing logarithms over GF(p) and its cryptographic significance
- Pohlig, Hellman
- 1978
(Show Context)
Citation Context ...All extension fields are chosen in such a way that the order of the group of points over the given extension field has a large prime factor. Therefore neither the algorithm of Pohlig and Hellman (see =-=[11]-=-) nor the attacks on supersingular curves (see [8]) can be used to compute discrete logarithms in the group of points on the given elliptic curves. We start with a short introduction to elliptic curve... |

284 |
Elliptic curve public key cryptosystems
- Menezes
- 1993
(Show Context)
Citation Context ...n the given elliptic curves. We start with a short introduction to elliptic curves, more information on elliptic curves over fields of characteristic two and their use in cryptography can be found in =-=[9]. Let IF q-=- be the finite field with q elements, where q is a "small" power of two, and let IF q be its algebraic closure. A (non supersingular) elliptic curve E over IF q can be defined by an equation... |

119 |
CM-curves with good cryptographic properties
- Koblitz
- 1991
(Show Context)
Citation Context ...The most time consuming operation of these cryptosystems is multiplication of a point on the elliptic curve with an integer, whose size is approximately equal to the order of the underlying group. In =-=[3], Nea-=-l Koblitz proposed to distinguish between the field of definition for the elliptic curve E and the field for the group of points on E. He suggested to use the group of points on so called "anomal... |

99 | Fast Key Exchange with Elliptic Curve Systems
- Schroeppel, Orman, et al.
- 1995
(Show Context)
Citation Context ...cal reasons, elliptic curves over fields of characteristic two are of special interest. Diffie-Hellman type cryptosystems using elliptic curves over IF 2 155 were implemented and compared to RSA (see =-=[12]-=-). The most time consuming operation of these cryptosystems is multiplication of a point on the elliptic curve with an integer, whose size is approximately equal to the order of the underlying group. ... |

83 | More flexible exponentiation with precomputation
- Lim, Lee
- 1994
(Show Context)
Citation Context ...0 1 4 log 2 (m) + 144 3 1 3 log 2 (m) + 61 2 9 log 2 (m) + 363 1 6 log 2 (m) + 2455 4 1 4 log 2 (m) + 311 1 6 log 2 (m) + 3279 1 8 log 2 (m) + 41759 10 4.2 Several Multiplications of a Fixed Point In =-=[6]-=-, the authors present a fast exponentiation techniques which reduces the number of multiplications if we want to exponentiate a fixed element several times. We can extend these ideas to our situation.... |

45 |
Reducing elliptic curve logarithms to a finite field
- Menezes, Okamoto, et al.
(Show Context)
Citation Context ...the order of the group of points over the given extension field has a large prime factor. Therefore neither the algorithm of Pohlig and Hellman (see [11]) nor the attacks on supersingular curves (see =-=[8]-=-) can be used to compute discrete logarithms in the group of points on the given elliptic curves. We start with a short introduction to elliptic curves, more information on elliptic curves over fields... |

18 |
Efficient Multiplication on Certain Nonsupersingular Elliptic Curves
- Meier, Staffelbach
- 1993
(Show Context)
Citation Context ...he field of definition for the elliptic curve E and the field for the group of points on E. He suggested to use the group of points on so called "anomalous" elliptic curves for such cryptosy=-=stems. In [7]-=-, the authors showed how to speed up point multiplication on the anomalous curve y 2 + yx = x 3 + 1, defined over IF 2 . In this paper, we extend the ideas presented in [7] to arbitrary non supersingu... |

4 |
A C++ library for computational number theory. Available at http://www.informatik.tu-darmstadt.de/TI/LiDIA/. Damien Stehlé
- LIDIA
(Show Context)
Citation Context ... We present some running times achieved with our implementation of Algorithm 6. As underlying field arithmetic, we use an implementation written by Patric Kirsch (see [2], also included in LiDIA, see =-=[5]-=-). This implementation uses a polynomial basis representation for finite fields of characteristic two, where the irreducible modulus is chosen as sparse as possible. For comparison, we first list aver... |

1 |
Rohm: Effiziente Digitale Signatursysteme auf der Basis elliptischer Kurven
- Fox, W
- 1996
(Show Context)
Citation Context ... in Section 5. 4 Variants of Algorithm 6 4.1 The Block Variant Several authors have used block techniques for speeding up the ordinary binary method for multiplication of points (for an overview, see =-=[1]-=-). We can use similar techniques to develop a Block Frobenius expansion algorithm. Again, the first part of the block version consists of computing a Frobenius expansion for the given integer m. Inste... |

1 |
Implementierung einer Arithmetik des Polynomrings GF(2)[X] und des
- Kirsch
- 1996
(Show Context)
Citation Context ... table will grow. 5 Running Times We present some running times achieved with our implementation of Algorithm 6. As underlying field arithmetic, we use an implementation written by Patric Kirsch (see =-=[2]-=-, also included in LiDIA, see [5]). This implementation uses a polynomial basis representation for finite fields of characteristic two, where the irreducible modulus is chosen as sparse as possible. F... |